Geology Reference
In-Depth Information
10
4
10
3
95% C.I.
10
2
10
1
-1.5
-1.0
-0.5 0.0
Frequency in cycles per year (cpy)
0.5
1.0
1.5
Figure 4.4 Spectral density of the VLBI polar motion based on the DFT found by
singular value decomposition (SVD).
discrete Fourier transform series of length 3513. The record segment length is then
26.25 years and the midpoint of the segment is at 17.125 years into the record. The
number of singular values eliminated (NSVE) to bring the Parseval ratio, expressed
by (2.202), as close as possible to unity, is found to be 320. The relative error in
reconstructing the sequence g
j
by the representation (4.12), or the relative recon-
struction error (RRE), is found to be 0.00081545722 for a Parseval ratio (PR) of
0.69568607.
To maximise frequency resolution we use the single segment of length
26.25 years. The spectral density estimate is then given by the normalised squared
magnitude of the discrete Fourier transform (2.350). The spectral density estimate
for the single segment is then χ
2
2
distributed with two degrees of freedom (2.355).
For two degrees of freedom, the probability density function (2.425) becomes
simply
e
−
x
/2
e
−
x
/2
. A fraction
α of the realisations will be below the random variable
x
for α
=
/2, and the cumulative distribution function is 1
−
e
−
x
/2
or
1
−
x
(α)
=−
2log(1
−
α). On logarithmic plots of the spectral density, the confidence
interval is the fixed length log
10
x
(1
−
α/2)
−
log
10
x
(α/2)
(2.355). For a 95% con-
fidence interval the fixed length is 0.8679
2.1634. The spectral density
estimate for the single segment of length 26.25 years is shown plotted in Figure 4.4.
+
1.2955
=
4.2.3 Interpolation of the VLBI pole path
In many applications, such as the use of the fast Fourier transform (FFT) to find the
discrete Fourier transform (DFT), or the application of maximum entropy spectral
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